The density distribution in the pre-Wolf-Rayet surroundings is directly responsible
for the shaping of our Wolf-Rayet nebulae and the stellar magnetic field seems to be dynamically
unimportant in the process of producing the bipolarities, in the sense
that only evolving MHD
flows in rotation produce spherical nebulae, at least in the corner of the parameter space of
blue supergiant and subsequent Wolf-Rayet stars that we explore.
An equatorially compressed circumstellar distribution gives the Wolf-Rayet stellar wind a
peanut-like morphology, which is further elongated along the polar direction as a jet-like
shape if the polar-to-equatorial density distribution ratio increases, or in case a
substantially long phase transition period (\(\sim \rm kyr\) ) between the two winds.
The topology of the magnetic field lines is greatly modified when the Wolf-Rayet wind
starts blowing into the previous cold material. The rotating blue supergiant
dipole \(B_{\phi }\)  sees its magnetic field lines opened and channelled to the poles of the structure.
Interestingly, all our nebulae are shaped by wind-wind interaction taking place in the
free-streaming stellar wind of the ancestor star, i.e. inside of the termination shock of
its main-sequence stellar wind bubble [1]}. This implies that the
nebulae will appear the same regardless of their driving star's bulk
motion [2]} and that the nebulae are free of ISM
material.
We propose to characterise the evolution of MHD asymmetric Wolf-Rayet nebulae in
the \(\rho \) -\(v\) -\(B_{\phi }\)  diagram, which we show to concisely report the main
feature of our MHD Wolf-Rayet nebulae as a characteristic S-shape.
